Math Help

The vertex points of a specific triangle define one, and only one, Euclidean (that is, flat) geometric plane. A specific Euclidean plane can, however, contain infinitely many different triangles. This is intuitively obvious when you give it a little thought. Just try to imagine three points that don’t lie in the same plane! Incidentally, this principle explains why a three-legged stool never wobbles. It is the reason why cameras and telescopes are commonly mounted on tripods (three-legged structures) rather than structures with four or more legs.

Sum of angle measures
In an triangle, the sum of the measures of the interior angles is 180 degrees (pi rad). This holds true regardless of whether it is an acute, right, or obtuse triangle, as long as all the angles are measured in the plane defined by the three vertices of the triangle.

Theorem of Pythagoras
Suppose we have a right triangle defined by points P, Q, and R whose sides are S, T, and U having lengths s, t, and u, respectively. Let u be the hypotenuse. Then the following equation is always true:

s² + r² = u²

The converse of this is also true: If there is a triangle whose sides have lengths s, t, and u, and the above equation is true, then that triangle is a right triangle.

Perimeter of a triangle
Suppose we have a triangle defined by points P, Q, and R, and having sides, S, T, and U of lengths s, t, and u. Then the perimeter, B, of the triangle is given by the following formula:

B = s + t + u

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The fundamental rules of geometry go all the way back to the time of the ancient Egyptians and Greeks, who used geometry to calculate the diameter of the earth and the distance to the moon. They employed the laws of Eculidean geometry (named after Euclid, a Greek mathematician who lived in the 3rd century BC). Euclidean plane geometry involves points and lines on perfectly flat surfaces.

In plane geometry, certain starting concepts aren’t defined formally, but are considered intuitively obvious. The point and the line ar examples. A point can be envisioned as an infinitely tiny sphere, having height, width, and depth all equal to zero, but nevertheless possessing a specific location. A line can be thought of as an infinitely thin, perfectly straight, infinitely long wire.

Naming points and lines
Points and lines are usually named using uppercase, italicized letters of the alphabet. The most common name for a point is P (for “point”), and the most common name for a line is L (for “line”). If multiple points are involved in a scenario, the letters immediately following P are used, for example Q, R, and S. If two or more lines exist in a scenario, the letters immediately following L are used, for example M and N. Alternatively, numeric subscripts can be used with P and L. Then we have points called P1, P2, P3, and so forth, and lines called L1, L2, L3, and so forth.

Two point principle
Suppose that P and Q are two different geometric points on a line. Two distinct points define one and only one line L. The following two statements are always true:

• P and Q lie on a common line L
• L is the only line on which both points lie

Distance notation
The distance between any two points P and Q, as measured from P towards Q along the straight line connecting them, is symbolized by writing PQ. Units of measurement such as meters, feet, millimeters, inches, miles, or kilometers are not important in pure mathematics, but they are important in physics and engineering. Sometimes a lowercase letter, such as d, is used to represent the distance between the points.

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